146 THE FASCINATION OF NUMBERS

totalling 34, may be made by taking any four num-
bers in square formation to each other, thus:

 

This property is not found in the first square;
(¢) the four central numbers in each square also total 34.

In every magic square the total of every row, etc., is
directly related to the number of integers used. It is quite
apparent, of course, that if the lowest integer used is 1 and all
the integers used are consecutive, then the highest integer
(and thus also the total number of integers used) must be a
square number of the form 72

The total obtained by adding together all the integers
used is of the form:

n?(n2+1)
2
being the sum of an arithmetical progression whose first
term is 1, whose last term is n2 and whose common difference
is 1. This total has to be spread equally over n rows (or,
looked at from another angle, over » columns) and the total
of each row or column is therefore:

n*(n®+1) n(n®41)

2n 2

In the two example squares shown, there are 16 consecu-
tive integers used. For this purpose, therefore, n2=16, and
n=4. The total of each row, column or diagonal is therefore

17

required to be %: 34, as shown.

There are a number of rules which enable one to con-
struct a magic square without much difficulty, and the rules